Inserted: 24 may 2010
Last Updated: 13 nov 2013
Journal: Proc. Roy. Soc. Edinburgh Sect. A
The relation between quasiconvexity and $k$-quasiconvexity, $k \geq 2$, is investigated. It is shown that every smooth strictly $k$-quasiconvex integrand with $p$-growth at infinity, $p>1$, is the restriction to $k$-th order symmetric tensors of a quasiconvex function with the same growth. When the smoothness condition is dropped, it is possible to prove an approximation result. As a consequence, lower semicontinuity results for $k$-th order variational problems are deduced as corollaries of well-known first order theorems. This generalizes a previous work by Dal Maso, Fonseca, Leoni and Morini, in which the case $k=2$ was treated.
Keywords: quasiconvexity, higher order variational problems